A sealed rectangular box measuring 8•6•18 contains 864 sugar cubes, each measuring 1•1•1. How many sugar cubes are touching the box?
Think about an 8*16*8 rectangular prism (made of 1*1*1 sugar cubes) that is empty on the interior. This is all the sugar cubes that are touching the box. To 'empty' the box, you would have to subtract a box slightly smaller in dimensions. Subtracting 2 from each dimension, you get a 6*14*6 box that would fit inside of the 8*16*8 box. (8*16*8)-(6*14*6) = 520 sugar cubes that are touching the box.
+129840 Let's look at this in a slightly different manner. Orient the box so that the base is 8 x 6 and the height is 18.
For the bottom layer, and top layers, 48 cubes are touching the box in each layer.
For the other 16 layers, 48 - 16 - 8 = 24 cubes are NOT touching the box in each layer.
So.......the number of cubes rouching the box must be the total cubes less the number that are not touching the box =
864 - 16(24) = 480 cubes touch the box
Another way to see this is that 96 cubes in total touch the top and bottom layers. And for each of the other 16 layers, as many cubes in each layer touch the box as the ones that don't
So 96 + 16(24) = 480 cubes touch the box
